## Abstract

Let w be a complex symmetric matrix of order r, and Δ _{1}(w), . . . , Δ _{r} (w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δ _{k} (w)/Δ _{k-1}(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.

Original language | English |
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Pages (from-to) | 697-711 |

Number of pages | 15 |

Journal | Mathematische Zeitschrift |

Volume | 259 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 2008 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)